Longitudinal studies are essential to research on growth, aging, and chronic disease, but longitudinal data are often underutilized because of difficulties in identifying and implementing appropriate statistical methods, and because the limitations of some popular methods are not recognized. In the proposed research, the investigators will continue to develop accessible methods for the analysis of longitudinal data, to create and share the software needed for such analyses, and to evaluate alternative methods, identifying those most appropriate for specifc analytic objectives. The research will emphasize methods that are easily implemented, efficient, and can accommodate unbalanced designs, missing observations, and subject attrition-common features of biomedical data. One focus of research will be two-stage methods, those that first summarize the data for each study unit and then analyze these summary statistics by univariate methods. Some two-stage methods may be biased as well as inefficient, but proper two- stage analyses can be efficient and appealing to medical investigators. Families of two-stage estimators will be studied to identify optimal procedures. Two-stage methods for nonlinear growth curve analysis will receive special attention. Optimal methods are often readily available for analyzing longitudinal data aset having neither missing observations nor dropouts. Analogues of these easily understood methods will be developed for incomplete data sets. Two approaches will be studied, one based on maximum likelihood using the EM algorithm and a simpler approach based on imputation of missing values. Work on repeated categorical response will focus on three families of models: marginal, transitional, and random effects. Two aproaches to marginal models will be pursued, those that explictly model only the occasion-specific distributions and maximum likelihood methods based on explicit multivariate distributions. Efficiency and sensitivity to bias will be compared for these two approaches. Work on transitional models will focus on identification of transitional models appropriate for characterization of life processes, such as the onset of chronic obstructive pulmonary disease or loss of functional autonomy with aging. The random effects model offers a parsimonious parametric alternative to general multivariate models. Exact maximum likelihood methods for fitting mixed models to repeated categorical observations will be developed and extended to accommodate incomplete data.